# Transforming ordinary linear differential equations to constant coefficient differential equations

## Abstract

The main purpose of this thesis is to give a characterization
of nth order linear homogeneous differential equations that can be
transformed into constant coefficient differential equations. The
characterization given makes use of invariance theory originated
by G. H. Halphen around 1880.
Chapter 1 gives an introduction to the problem of transforming
ordinary linear homogeneous differential equations into constant
coefficient differential equations. Chapter 1 includes an example
as well as enough of the theory of invariants to give a proof of
the main theorem of this thesis.
Chapter 3 is a development of the transform equations that we
make use of throughout this thesis. That is, we make changes of
the dependent variable and/or independent variable of a given
differential equation. The transformed equation is expressed in
terms of the coefficients of the original differential equation and
the functions used to define the changes of variables that we have
made.
In the last section of Chapter 4, we prove an important
invariance relation that is used in Chapter 5. The rest of Chapter
4 contains invariance results that are of historical interest as
well as being a prelude to the results of Chapter 5.
Chapter 5, the most important chapter of this thesis, concerns
the invariance theory of Halphen. This invariance theory is needed
to give the characterization of nth order differential equations that
can be transformed into constant coefficient differential equations.
Chapter 6 is devoted to applying the preceding theory to the
solution of differential equations that can be transformed into
constant coefficient differential equations.
The appendix contains some interchange of summation formulas
that we use throughout the thesis.